Degenerate Stochastic Systems and Related Problems in Analysis

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9703596 Mohammed The investigator will continue his collaborative work with Denis Bell on the study of degenerate stochastic differential equations and related problems in linear and quasilinear second-order partial differential equations. The proposed research falls into three parts. Part I deals with degenerate diffusions and their impact on linear partial differential equations. In their recent work, the investigators have proved a very general Hormander-type hypoellipticity theorem for second-order linear partial differential operators. The hypotheses of this theorem allow Hormander's general Lie algebra condition to fail at an optimal exponential rate on smooth hypersurfaces in Euclidean space. Such operators have been termed superdegenerate. The investigators will establish the existence of smooth solutions to the Dirichlet and Neumann problems associated with superdegenerate operators. In Part II, the investigators will study the existence of smooth densities for a wide class of degenerate stochastic hereditary equations. They will seek to use their methods to establish hypoellipticity of the corresponding operators. In addition to solving an infinite-dimensional hypoellipticity problem (apparently the first of its kind), the estimates obtained here should lead to the existence of a Lyapunov spectrum in probability for singular hereditary systems. In Part III, they will use their methods to study quasilinear second-order partial differential operators with superdegenerate principal parts. These operators are closely related to superdiffusions. The objective of this part of the research is to seek classical smooth positive solutions of the quasilinear Cauchy, Dirichlet, and Neumann problems associated with such operators. This research deals with two important problem areas that arise in physics and engineering. The first area concerns an important class of mathematical models, called partial differential equations, that are fundamental obje cts in modern day pure and applied mathematics. These equations arose from the study of heat conduction, electrical potential, and fluid flow. Partial differential equations have important connections with several areas of mathematics, in particular probability theory and geometry. The second area is devoted to a class of models that are used in physics, engineering and biology in order to analyze dynamical systems whose evolution is influenced by random fluctuations and past history. These models are very important in a variety of diverse areas ranging from signal processing, stock market fluctuations, economic and labor models, aircraft dynamics, materials with memory, and population dynamics. The investigators will use the most current probabilistic techniques in order to develop a deeper understanding of these models.